PRESENTED BYROMIKA CHOUDHARY

FRACTIONAL DIFFERENTIATORS USING

FIR AND IIR FILTERS

GUIDED BYMs. VINEETA

Contents

ObjectiveIntroduction to Fractional CalculusDigital differentiator Fractional DifferentialWhat does fractional derivative mean?Basic PrincipleFir Filter

Objective

The objective of this project is to use signal-processing tools such as the Fourier

transform and the digital FIR differentiator, to compute fractional derivatives of signals.

Fractional Calculus

Fractional Calculus is the branch of calculus that generalizes the derivative of a function to non-integer order, allowing calculations such as deriving a function to 1/2 order.

The name "fractional" is used for denoting the kind of derivative.

Mathematical analysis

The derivative of a function f is defined as

nth derivative of a function

h

xfhxfxfD

h

)()(lim)(

0

1

What does fractional derivative mean?

(Non integer order differentiator)

The Fourier transform of differentiation of the function f(x)

F[D{f(x)}]=(jw)F(w)

where F(w) is the Fourier transform of f(x). The nth derivative of f(x)

F[Dn {f(x)}]=(jw)n F(w) ; let the initial value (f(0-))=0

If n is a fractional order (suppose n=v)

F[Dv {f(x)}]=(jw)v F(w)

“It means that a fractional order differentiation is the output of a filter whose frequency response is (jw)v ”

FIR Fractional derivatives

To compute the fractional derivative we have to design an FIR filter having frequency response (jw)v

,0,

,0,0

,0,

)()(2/

2/

wew

w

wew

iwwDivv

ivv

v

Since the FIR filter will introduce a delay and D(w) is a zero delay specification, the FIR filter cannot be used to approximate the ideal response D(w). That’s why we have to introduce a delay system in cascade with this differentiator. Thus

where n0 is a prescribed delay.

,)()( 0wind ewDwF

Once filter (H(w)) is designed, the following formula is used to obtain the design result of the fractional differentiator:

Now by passing any signal through this filter we can compute the fractional derivative of that signal.

winewHwD 0)()(ˆ

IIR Fractional derivatives

the discretization of the fractional-order differentiator sr (r is a real number) can be expressed by the so-called generating function s=w(z). This generating function and its expansion determine both the form of the approximation and the coefficients.The DFOD is then obtained by using the CFE or a new recursive expansion formula.

By using the trapezoidal rule the generating function can be written as:

Thus the final expression for DFDO will be:

22

22

221

2

))(3(

)1(6

))(3(

)1(6

)(~1

)(rzaT

zr

rzaTr

z

zHzw

r

r

bz

zkzwzG

21

2

011

)1(

1))(()(

Conclusion

The FIR fractional order differentiator can be used to generate a random fractal process which is better than the process obtained by the conventional method. However, using an FIR filter to approximate (iw)v may be less efficient due to the very high order of the FIR filter. The IIR fractional differentiator is having a tuning knob to compromise the high frequency approximation accuracy.

THANKS